Let (Y, g) denote a compact, oriented Riemann 3-manifold without boundary. Denote by π : X → Y a principal S1-bundle over Y , and by Z → Y the associated 2-disk bundle. Set c := c1(Z) ∈ H(Y,Z). Denote by tZ ∈ H2(Z, X;Z) the Thom class of Z → Y , by j the inclusion X ↪→ Z and by ζ : Y ↪→ Z the natural inclusion. Using the Thom isomorphism H•(Z) ∪tZ −→ H•+2(Z,X;Z), c = ζ∗tZ , and the long exact c...

We give the Thom polynomials for the singularities I2,2, and A3 associated with maps (C•, 0) → (C, 0) with parameter k ≥ 0. We give the Schur function expansions of these Thom polynomials. Moreover, for the singularities Ai (with any parameter k ≥ 0) we analyze the “first approximation” F (i) to the Thom polynomial. Our computations combine the characterization of Thom polynomials via the “meth...

The Thom-Boardman symbol was first introduced by Thom in 1956 to classify singularities of differentiable maps. It was later generalized by Boardman to a more general setting. Although the Thom-Boardman symbol is realized by a sequence of non-increasing, nonnegative integers, to compute those numbers is, in general, extremely difficult. In the case of polynomial multiplication maps, Robert Varl...

We generalize the notion of Thom polynomials from singularities of maps between two complex manifolds to invariant cones in representations, and collections of vector bundles. We prove that the generalized Thom polynomials, expanded in the products of Schur functions of the bundles, have nonnegative coefficients. For classical Thom polynomials associated with maps of complex manifolds, this giv...

Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry (enumerative geometry, moduli spaces) and algebraic combinatorics. The main obstacle of their widespread application is that only a few, sporadic Thom polynomials have been known explicitly. In this...

We apply an announced result of Blumberg-Cohen-Schlichtkrull to reprove (under restricted hypotheses) a theorem of Mahowald: the connective real and complex K-theory spectra are not Thom spectra. The construction of various bordism theories as Thom spectra served as a motivating example for the development of highly structured ring spectra. Various other examples of Thom spectra followed; for i...

I t has been known for some time that Penicillium citrinum Thom when grown on suitable media produces a characteristic lemon yellow colour. Thus Thom in his monograph on “ Cultural Studies of species of 1910), says, “ P. citrinum produces a lemon yellow colour, soluble in alcohol, in media containing sugars, milk, gelatin, bread and potato.” This colouring matter was not isolated, however, nor ...

We present a formula describing the action of a generalised Steenrod operation of Z2-type [14] on the cohomology class represented by a proper self-transverse immersion f : M X. Our formula depends only on the Umkehr map, the characteristic classes of the normal bundle, and the class represented by the double point immersion of f . This generalises a classical result of R. Thom [13]: If α ∈ H(X...

Recall that a String manifold is a Spin manifold M together with a trivialization of the class usually denoted p1(M)/2; this is a characteristic class, defined only for Spin manifolds, so that twice it is the usual first Pontrjagin class. The bordism spectrum of String manifolds is called MString or MO〈8〉; it is the Thom spectrum of the map BO〈8〉 −→ BO of the 7-connected cover of BO to BO. Simi...

The notion of cobordism is simple; two manifolds M and N are said to be cobordant if their disjoint union is the boundary of some other manifold. Given the extreme difficulty of the classification of manifolds it would seem very unlikely that much progress could be made in classifying manifolds up to cobordism. However, René Thom, in his remarkable, if unreadable, 1954 paper Quelques propriétés...